Absolute Continuity of Monotone Aggregations under Positive Regression Dependence
Abstract
In this paper, we provide sufficient conditions for the absolute continuity of one-dimensional push-forwards of dependent random vectors. Suppose that X has an absolutely continuous distribution and that the conditional distribution of an ℝd-valued random vector Y given X=x is nondecreasing in x∈ ℝ in the usual stochastic order. For Borel maps g: ℝ×ℝd→ℝ satisfying a coordinatewise monotonicity condition in Y and a uniform lower-increment condition in X, we prove that g(X,Y) has an absolutely continuous distribution. The result requires neither independence nor a joint density, and allows the marginal law of Y to be completely arbitrary. Moreover, the result remains valid if ℝd is replaced by an arbitrary measurable space endowed with a reflexive binary relation. We discuss consequences for monotone risk aggregation and extensions of the familiar regularization by convolution beyond independent random variables.
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BibTeX
@misc{goldys2026absolute,
title = {{Absolute Continuity of Monotone Aggregations under Positive Regression Dependence}},
author = {Ben Goldys and Max Nendel},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2606.21042},
eprint = {2606.21042},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2606.21042},
}
RIS
TY - UNPB
TI - Absolute Continuity of Monotone Aggregations under Positive Regression Dependence
AU - Ben Goldys
AU - Max Nendel
PY - 2026
DA - 2026/11/05
PB - arXiv
DO - 10.48550/arxiv.2606.21042
UR - https://arxiv.org/abs/2606.21042
AB - In this paper, we provide sufficient conditions for the absolute continuity of one-dimensional push-forwards of dependent random vectors. Suppose that X has an absolutely continuous distribution and that the conditional distribution of an ℝ^d-valued random vector Y given X=x is nondecreasing in x∈ ℝ in the usual stochastic order. For Borel maps g: ℝ×ℝ^d→ℝ satisfying a coordinatewise monotonicity condition in Y and a uniform lower-increment condition in X, we prove that g(X,Y) has an absolutely continuous distribution. The result requires neither independence nor a joint density, and allows the marginal law of Y to be completely arbitrary. Moreover, the result remains valid if ℝ^d is replaced by an arbitrary measurable space endowed with a reflexive binary relation. We discuss consequences for monotone risk aggregation and extensions of the familiar regularization by convolution beyond independent random variables.
ER -